• Exploiting problem structure: The used interior point methods are based on the solution of least squares problems, which can take the problem structure into account.
There are also many solver programs available, a few of them – the ones in particular that are free and can be used with MATLAB – are SeDuMi, SDPT3, MOSEK, and SDPA. A short description and a comprehensive benchmark of these and other solvers can be found in [46].
A.3 Group theory
This section follows the appendix of [5], concentrating on the most needed concepts.
A.3.1 General concepts
Definition A.4. A group(G,·) is a non-empty set G with a binary operation
“·” having the following properties:
• Closure: g1·g2 ∈G for all g1, g2 ∈G,
• Associativity: (g1·g2)·g3 =g1·(g2·g3) for all g1, g2, g3 ∈G,
• Identity: there exists e∈G such that ∀g ∈G, g·e=e·g =g,
• Inverse: for allg ∈G, there existsg−1 ∈Gsuch that g·g−1 =g−1·g =e.
We often leave out the operation · in g1·g2 and write simply g1g2.
A group G is finite if the number of elements, i.e., the order of G denoted by |G| is finite. A group G is Abelian if the operation · is commutative, i.e, g1g2 =g2g1 for all g ∈G.
A subgroup H of G is a subset of G which forms a group under the same operation · as G. In notation we write this as H < G. If g1, g2 ∈G then the conjugate of g2 with respect to g1 is g1g2g1−1 ∈ G. Now we can present two important notions.
Definition A.5. The centralizer Z(S) of a subset S ⊂ G (not necessarily a subgroup) is the set {g ∈G|gsg−1 =s,∀s ∈S}.
Definition A.6. The normalizer N(S) of a subset S ⊂ G (not necessarily a subgroup) is the set {g ∈G|gsg−1 ∈S,∀s∈S}.
The study of a group can be greatly simplified by the use of a special subset of the group as a compact description:
Definition A.7. The elementsg1, . . . , gl in a groupGare said to be the genera-tors ofGif every element ofGcan be written as a product of (possibly repeated) elements from the set {g1, . . . , gl}. In notation we write G=hg1, . . . , gli.
It can be shown that a group G can always be generated with a set of at most log2(|G|)independent generators.
The following definition is of essential importance in the thesis:
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Definition A.8. For a subgroup H < G, the left coset of H in G determined byg ∈Gis the setgH :={gh|h∈H}. The right cosetHgis defined similarly.
Elements of a particular coset gH are known as coset representatives of that coset. CosetsgH define an equivalence relation∼ onG given by g1 ∼g2
if and only if g1h=g2 for some h∈H. It follows that any two left cosets gH inG are either identical or disjoint. In other words the set of left cosetsG/H form a partition of G.
An example of groups is the group ofn×nunitary matricesU(n)with the matrix multiplication as binary operation. Another example important in the thesis is the Pauli group onnqubitsPn< U(2n). It consists of alln-fold tensor products of the Pauli matrices defined in (2.2) with a possible±1or±ifactor.
Thus it is generated as Pn = hSn
i=1{Xi, Zi},i1i, that is, all Pauli operators Xi,Yi, andZi acting only on theith qubit together with the possible±1or±i factor. The order of the Pauli group is |Pn| = 22n+2. For example, the Pauli group for one qubit is P1 = hX, Z,i1i. The most important properties of Pn are the following:
1. All g ∈ Pn are either Hermitian or antihermitian.
2. Any two elements g, h ∈ Pn either commute or anticommute, i.e., gh =
±hg. Note that this property implies that for any subset S ⊂ Pn for which −1 ∈/ S, the centralizer Z(S) and the normalizer N(S) are the same.
In many cases group elements g ∈ G can be tought of as transformations on some other setV. In this context we can speak about stabilizers.
Definition A.9. The group element g ∈G fixes (stabilizes) x∈ V if gx =x.
For any x∈ V the stabilizer subgroup S < G of x is the set of all g ∈ G that fix x, i.e., S ={g ∈G|gx=x}.
In particular, ifV is a vector space then it is easy to see that the elements of G stabilize their whole +1 eigenspace. The applications of this property in quantum information processing is discussed in the followig using the Pauli groupPn.
A.3.2 The stabilizer formalism
The stabilizer formalism is an advantageous group theoretic approach to a wide class of actions in quantum mechanics with main application in quantum error correction (see in appendix A.5.2). The presentation follows [5].
Stabilizers of quantum states
Let S be a subgroup of the Pauli group Pn on n qubits. Then the space VS is the space of alln-qubit states fixed by the elements ofS, i.e., the space obtained as the intersection of +1 eigenspaces of all g ∈ S. In other words VS is the vector space stabilized by the stabilizer S: g|ψi = |ψi, ∀|ψi ∈ VS,
∀g ∈S.
A.3. Group theory
A big advantage of the stabilizer formalism comes from the possibility of describing a group by its generators. A vector is stabilized by S if and only if it is stabilized by the generators ofS. The fact that any groupG has at most log2(|G|) generators allows a very compact representation and easy handling of stabilizers. For example an n-qubit state is stabilized by a subgroup of Pn having n Pauli group generators and a generator is a product of the at most 2n+ 2number of original Pauli group generators. This means that all states of VS can be described by at most O(n2)amount of information.
In practice we seek the stabilizer of nontrivial vector spaces. It can be seen that the vector space VS is nontrivial if and only if all element of S commute and −1 ∈/ S. Thus in the following we always assume independent generators having these properties. For nontrivial VS the following statements hold:
1. If the subgroupS < Pn has n−k generators thenVS has 2k dimensions.
2. For the stabilizer generators of every S =hg1, . . . , glithere exist g ∈ Pn such that ggig† =−gi for some i, but ggjg† =gj, ∀j 6=i.
Unitary transformations using stabilizers
The stabilizer formalism can also be used to describe unitary dynamics in vector spaces. Suppose we act with a unitary U on the vector space VS stabilized by S <Pn. Let |ψi ∈ VS. Then for all g ∈S
U|ψi=U g|ψi=U gU∗U|ψi , (A.4) thus the state U|ψi is stabilized by U gU∗. It follows that the stabilizer of UVS is the group U SU∗ ≡ {U gU∗|g ∈S}. Furthermore, if S is generated by g1, . . . , gl then U SU∗ is generated by U g1U∗, . . . , U glU∗. Thus it is enough to compute the effect of U on the generators.
Note that using these principles, the stabilizer formalism allows an effi-cient classical simulation of many quantum behaviour. However, it can not efficiently simulate all of quantum mechanics. The exact limitations are stated by the famous Gottesman–Knill theorem [5]. In general those unitaries (gates) U can be efficiently simulated, for which UPnU∗ = Pn, i.e., elements of the normalizer N(Pn). Fortunately, encoding, decoding, error-detection and re-covery for stabilizer based quantum error correcting codes (see Chapter 3) are all such normalizer gates.
Quantum measurement using stabilizers
Using the stabilizer formalism, measurements made in the computational basis can also be efficiently described. Apart from a possible −1,±i factor, g ∈ Pn is self-adjoint and can be seen as an observable. Suppose then g is an observable and the system is in state|ψiwith stabilizer S=hg1, . . . , gli. Then there are two possibilities:
1. g commutes with all generators of S.
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2. g anticommutes with g1 and commutes with all other gi, i > 1. This is not a loss of generality, because if g anticommutes also with gi then it commutes with g1gi, thus we can just swap gi with g1gi.
In the first case gjg|ψi = ggj|ψi = g|ψi implies g|ψi ∈ VS, i.e., g|ψi is proportional to|ψi, because VS one dimenisonal. The hermiticity ofg implies g2 = 1, thus g|ψi = ±|ψi. This means that either g or −g is element of S.
If g ∈ S then g|ψi= |ψi and the measurement result is 1 with 1 probability.
If in turn −g ∈ S then g|ψi = −|ψi and the measurement result is −1 with 1probability. Furthermore, the measurement does not ruin the state, because the stabilizer does not change.
In the second case, as the eigenvalues ofgare±1the spectral decomposition ofg isg =P+−P−, where P+ = 1+g2 is the projection onto the+1 eigenspace and P− = 1−2g is the projection onto the −1 eigenspace. Then using the measurement postulate together withgg1 =−g1g the p+ and p− probabilities of the +1 and −1 results:
p+ =hψ|P+g1|ψi=hψ|g1P−|ψi=p−= 1 2 . Then the state after measurement will be
|ψ+i=√
2P+|ψi, with stabilizer hg, g2, . . . , gli ,
|ψ−i=√
2P−|ψi, with stabilizer h−g, g2, . . . , gli .